Cube sums of form $3p$ and $3p^2$

Jie Shu; Xu Song; Hongbo Yin

Cube sums of form $3p$ and $3p^2$

Number Theory 2018-04-11 v1

Authors: Jie Shu , Xu Song , Hongbo Yin

Abstract

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton-Dyer conjecture of the related elliptic curves.

Keywords

cube sum Heegner point Gross-Zagier formula elliptic curve Birch and Swinnerton-Dyer conjecture prime

Cite

@article{arxiv.1804.02924,
  title  = {Cube sums of form $3p$ and $3p^2$},
  author = {Jie Shu and Xu Song and Hongbo Yin},
  journal= {arXiv preprint arXiv:1804.02924},
  year   = {2018}
}

Comments

arXiv admin note: text overlap with arXiv:1708.05266

Cube sums of form $3p$ and $3p^2$

Abstract

Keywords

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